On the Subspace Orbit Problem and the Simultaneous Skolem Problem

The Orbit Problem asks whether the orbit of a point under a matrix reaches a given target set. When the target is a single point, the problem was shown to be decidable in polynomial time by Kannan and Lipton. This decidability result was later extended by Chonev et al. to targets of dimension 3 (in arbitrary ambient dimension), but decidability remains open for subspaces of dimension 4. At the other extreme, the special case of the Orbit Problem in which the target set is a hyperplane of co-dimension 1 is equivalent to the Skolem Problem for linear recurrence sequences, whose decidability has been open for many decades.

In this paper, we show that the Orbit Problem is decidable if the target subspace has dimension logarithmic in the dimension of the orbit. Over the rationals, we moreover obtain a complexity bound NP RP in this case, when the target space dimension is bounded. On the other hand, we show that the version of the Orbit Problem where the dimension of the target subspace is linear in the dimension of the orbit is as hard as the Skolem Problem.